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14.2.1 Derivation for F = ma in 3D<br />
F = d dt p<br />
= d dt (mv)<br />
= m d dt v<br />
⎡⎛<br />
⎞ ⎛ ⎞ ⎛ ⎞⎤<br />
a x<br />
ω x<br />
v x<br />
= m<br />
⎢⎜a y ⎟<br />
⎣⎝<br />
⎠ + ⎜ω y ⎟<br />
⎝ ⎠ ⊗ ⎜v y ⎟⎥<br />
⎝ ⎠⎦<br />
a z ω z v z<br />
⎡⎛<br />
⎞ ∣ ∣⎤<br />
∣∣∣∣∣∣∣∣∣ ∣∣∣∣∣∣∣∣∣ a x<br />
i j k<br />
= m<br />
⎢⎜a y ⎟<br />
⎣⎝<br />
⎠ + det ω x ω y ω z ⎥<br />
⎦<br />
a z v x v y v z<br />
⎡⎛<br />
⎞ ⎛<br />
⎞⎤<br />
a x<br />
ω y v z − ω z v y<br />
= m<br />
⎢⎜a y ⎟<br />
⎣⎝<br />
⎠ + ⎜− (ω x v z − ω z v x )<br />
⎟⎥<br />
⎝<br />
⎠⎦<br />
a z ω x v y − ω y v x<br />
⎛<br />
⎞<br />
a x + ω y v z − ω z v y<br />
= m<br />
⎜a y − ω x v z + ω z v x ⎟<br />
⎝<br />
⎠<br />
a z + ω x v y − ω y v x<br />
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